The long-running dispute over the center had thrown a pall over ongoing negotiations for a bilateral security agreement to govern the presence of U.S. forces in Afghanistan after 2014.
An initial agreement to hand over Parwan was signed a year ago, but efforts to follow through on it constantly stumbled over American concerns that the Afghan government would release prisoners that it considered dangerous. An initial deadline for the full handover passed last September; another passed earlier this month.
The U.S. concerns are not without foundation. Zakir Qayyum, a former Guantanamo detainee, was released into Afghan custody in 2007. Freed four months later, he rejoined the Taliban and reportedly has risen to become the No. 2 leader in the Taliban.
Both Kerry and Karzai lauded the transfer of the facility. Karzai said an Afghan review board would carefully consider any intelligence provided by the U.S. or others about detainees they deem to be too dangerous to free.
The pair also called on the Taliban to take advantage of the offer to open a political office in Doha, Qatar, from where they could engage in reconciliation talks with the Afghan Peace Council and potentially negotiate an end to hostilities.
Kerry said the Taliban should not ignore the opportunity because the United States is committed to Afghanistan's security beyond 2014 and will not allow gains made over the past decade to be lost. He noted that Obama has not yet decided how many U.S. troops should stay after next year and that the Taliban should not count on a complete American withdrawal.
There are about 100,000 coalition troops in Afghanistan, including about 66,000 from the United States. Although there is no decision on a residual force, U.S. officials have said as many as 12,000 U.S. and coalition forces could remain.
Karzai said that peace talks with the Taliban would require the involvement of Pakistan because any Afghan peace process without that country was doomed to failure. Pakistan, particularly its intelligence service, has close ties to members of the Taliban.
Kerry, who arrived in Kabul from Amman, Jordan, had hoped to travel to Pakistan on this trip to the region but put it off due to elections there.
Instead, he met late Sunday in Amman with Pakistani army chief for Gen. Ashfaq Parvez Kayani, U.S. officials said.
Kerry and Kayani had a private dinner at the residence of the U.S. ambassador to Jordan as Pakistan continued to seethe in the aftermath of the return of former president Pervez Musharraf, himself a former army chief, from exile.
In Kabul, Kerry planned to meet again Tuesday with Karzai. He also had meetings scheduled with civic leaders and others to discuss continued U.S. assistance to the country and how to wean it from such aid as the international military operation winds down. Upcoming national elections also were on his agenda.
Kerry praised what he said was Afghanistan's commitment to "safe, secure" and transparent elections next year that will see a successor to Karzai voted into power.
Patrick Quinn in Kabul and Rahim Faiez in Bagram, Afghanistan contributed to this report.